Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in the set of lower bounds for $A$.

Exercise: Suppose that $X$ is a poset (and thus also a preorder). Show that meets and joins in a poset are unique if they exist.

1 Answer
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Let $A \subseteq X$ and suppose $x$ and $y$ are both joins of $A$. Then $x$ and $y$ are both upper bounds of $A$. So $A \le x$ and $A \le y$. But since $x$ and $y$ are joins of $A$ and $x, y \in X$, $y \le x$ and $x \le y$. Therefore, $x = y$ because because $\le$ is anti-symmetric. Meets are unique by a similar argument.

Your solution seems fine. Just a remark about notation - I am not sure whether $A\le x$ is frequently used as a shortcut for $(\forall a\in A) a\le x$. (But if it used this way in your course/book, then it is certainly ok for you to use this notation.)
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Martin SleziakJul 24 '12 at 10:44

@MartinSleziak That is the notation used in the textbook I am reading. When i typed this up, I wondered if it was a standard notation or not.
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Code-GuruJul 24 '12 at 13:25